What are imaginary numbers?

Correct statement: "*i is equivalent to a 90° rotation in some contexts." Which is kind of obvious to anybody who has done some maths or physics at some point?

One great conclusion from this approach is how intuitive it becomes to understand the square root of i. I always thought you'd need another dimension to describe that, and another dimension for the square root of that unit, and so on.

But think of i as a 90° rotation, so applying it twice (squaring it) results in 180° which is -1. Then √i is a 45° rotation so applying it twice results in 90° which is i. Sure enough, this works out. The unit vector at 45° is 0.5√2 + 0.5√2 * i. Follow the rules of complex arithmetic to square that and you do indeed get i.

A rotation of -135° applied twice gives the same result. Square the unit vector of -0.5√2 + -0.5√2 * i and you also get i. We've arrived back at the axiom that all numbers have two square roots of opposite signs.

Last question: What's the cube root of i? Easy: a 30° rotation. The 30° unit vector is 0.5√3 + 0.5i, and cubing that does indeed get you i.

While I like the geometric content of the rest of your post, I have to quibble on two points:

    * that a number has two (usually distinct) square roots is a theorem, not an axiom.  That is, it's not 'built in' (unlike, say, commutativity of addition), but rather can be proven on the basis of 'built-in' properties.

    * while one may technically say that i and -i have 'opposite signs', it's a good idea not to talk about the sign of a complex number at all (except possibly in the sense that sgn(z) = z/|z|).  It's better to say that the two square roots are opposite; or, if one wants to be more precise, that they are additive inverses.